Data-Driven Self-Supervised Learning for the Discovery of Solution Singularity for Partial Differential Equations

TL;DR

This paper introduces a data-driven, self-supervised learning framework to detect unknown singularities in solutions of PDEs, improving computational efficiency in scientific computing.

Contribution

It proposes a novel SSL method with filtering techniques for singularity detection using only raw data, addressing challenges of data noise and unknown singularity locations.

Findings

Effective in detecting various singularities such as interior circles and boundary layers.

Robust against input perturbations and label corruption.

Demonstrates potential pathological issues with raw data without filtering.

Abstract

The appearance of singularities in the function of interest constitutes a fundamental challenge in scientific computing. It can significantly undermine the effectiveness of numerical schemes for function approximation, numerical integration, and the solution of partial differential equations (PDEs), etc. The problem becomes more sophisticated if the location of the singularity is unknown, which is often encountered in solving PDEs. Detecting the singularity is therefore critical for developing efficient adaptive methods to reduce computational costs in various applications. In this paper, we consider singularity detection in a purely data-driven setting. Namely, the input only contains given data, such as the vertex set from a mesh. To overcome the limitation of the raw unlabeled data, we propose a self-supervised learning (SSL) framework for estimating the location of the singularity. A key component is a filtering procedure as the pretext task in SSL, where two filtering methods are presented, based on $k$ nearest neighbors and kernel density estimation, respectively. We provide numerical examples to illustrate the potential pathological or inaccurate results due to the use of raw data without filtering. Various experiments are presented to demonstrate the ability of the proposed approach to deal with input perturbation, label corruption, and different kinds of singularities such interior circle, boundary layer, concentric semicircles, etc.